Length functions and Demazure operators for G(e,1,n), II

This note is the first part of consecutive two papers concerning with a length function and Demazure operators for the complex reflection group W = G(e, 1, n). In this first part, we study the word problem on W based on the work of Bremke and Malle BM]. We show that the usual length function ?(W) associated to a given generator set S is completely described by the function n(W), introduced in BM], associated to the root system of W.In the second part, we will study the Demazure operators of W on the symmetric algebra. We define a graded space H_W in terms of Demazure operators, and show that H_W is isomorphic to the coinvariant algebra S_W, which enables us to define a homogeneous basis on S_W parametrized by w?W.